Oceanside Spider House Code Guessing

Oceanside Spider House Mask Code Guessing

There are 4*3^5 = 972 possible codes.

Note that I assume a uniform probability distribution (i.e. every mask has an equal chance of being the correct choice).

The Mean number of arrows needed when guessing is 27.5
The Minimum number of arrows needed when guessing is 6
The Maximum number of arrows needed when guessing is 49

It is notable that you can guess the correct mask code with 30 or fewer arrows with 624/972 probability which is approximately 64.2%. (Find this in the list below with Ctrl + F "30 or less")

Below is a list of 2-tuples. The first entry in the list is ('6 or less', '1 out of 972 Probability') which means that with 1/972 probability you can guess the correct mask code with 6 or fewer arrows.


[('6 or less', '1 out of 972 Probability'),
 ('7 or less', '2 out of 972 Probability'),
 ('8 or less', '4 out of 972 Probability'),
 ('9 or less', '7 out of 972 Probability'),
 ('10 or less', '11 out of 972 Probability'),
 ('11 or less', '17 out of 972 Probability'),
 ('12 or less', '25 out of 972 Probability'),
 ('13 or less', '35 out of 972 Probability'),
 ('14 or less', '48 out of 972 Probability'),
 ('15 or less', '63 out of 972 Probability'),
 ('16 or less', '82 out of 972 Probability'),
 ('17 or less', '104 out of 972 Probability'),
 ('18 or less', '129 out of 972 Probability'),
 ('19 or less', '158 out of 972 Probability'),
 ('20 or less', '190 out of 972 Probability'),
 ('21 or less', '225 out of 972 Probability'),
 ('22 or less', '264 out of 972 Probability'),
 ('23 or less', '305 out of 972 Probability'),
 ('24 or less', '348 out of 972 Probability'),
 ('25 or less', '393 out of 972 Probability'),
 ('26 or less', '439 out of 972 Probability'),
 ('27 or less', '486 out of 972 Probability'),
 ('28 or less', '533 out of 972 Probability'),
 ('29 or less', '579 out of 972 Probability'),
 ('30 or less', '624 out of 972 Probability'),
 ('31 or less', '667 out of 972 Probability'),
 ('32 or less', '708 out of 972 Probability'),
 ('33 or less', '747 out of 972 Probability'),
 ('34 or less', '782 out of 972 Probability'),
 ('35 or less', '814 out of 972 Probability'),
 ('36 or less', '843 out of 972 Probability'),
 ('37 or less', '868 out of 972 Probability'),
 ('38 or less', '890 out of 972 Probability'),
 ('39 or less', '909 out of 972 Probability'),
 ('40 or less', '924 out of 972 Probability'),
 ('41 or less', '937 out of 972 Probability'),
 ('42 or less', '947 out of 972 Probability'),
 ('43 or less', '955 out of 972 Probability'),
 ('44 or less', '961 out of 972 Probability'),
 ('45 or less', '965 out of 972 Probability'),
 ('46 or less', '968 out of 972 Probability'),
 ('47 or less', '970 out of 972 Probability'),
 ('48 or less', '971 out of 972 Probability'),
 ('49 or less', '972 out of 972 Probability')]


The number of arrows needed, N, can be thought of as a random variable N = W1 + W2 + W3 + W4 + W5 + W6

Generating functions (Gf's):
(Let Ui(s) be the Gf for Wi)

U1(s) = (1/4)s + (1/4)s^2 + (1/4)s^3 + (1/4)s^4
U2(s) = (1/3)s + (1/3)S^3 + (1/3)s^5
U3(s) = (1/3)s + (1/3)s^4 + (1/3)s^7
U4(s) = (1/3)s + (1/3)s^5 + (1/3)s^9
U5(s) = (1/3)s + (1/3)s^6 + (1/3)s^11
U6(s) = (1/3)s + (1/3)s^7 + (1/3)s^13